Tight bounds for expected propagation time of probabilistic zero forcing

TL;DR

This paper establishes tight bounds on the expected propagation time of probabilistic zero forcing in connected graphs, confirming conjectures and improving previous bounds.

Contribution

It proves the existence of initial sets with minimal size for optimal expected propagation time and tight bounds on the probabilistic throttling number.

Findings

Expected propagation time is approximately n/2 for a single initial vertex.

Initial sets of size O(√n) achieve expected propagation time of O(√n).

Results confirm conjectures by Narayanan and Sun, and improve previous bounds.

Abstract

We study the probabilistic zero forcing process, a probabilistic variant of the classical zero forcing process. We show that for every connected graph $G$ on $n$ vertices, there exists an initial set consisting of a single vertex such that the expected propagation time is $n/2 + O(1)$. This result is tight and confirms a conjecture posed by Narayanan and Sun. Additionally, we show tight bounds on the probabilistic throttling number, which captures the trade-off between the size of the initial set and the speed of propagation. Namely, we show that for every connected graph $G$ on $n$ vertices, there exists an initial set consisting of $O(\sqrt{n})$ vertices such that the expected propagation time is $O(\sqrt{n})$. This improves upon previous results by Geneson and Hogben, and confirms another conjecture by Narayanan and Sun.